Polyhedral annular structures and blanks for forming same

ABSTRACT

Folded structures which are polyhedrons of generally toroidal shape, movable to various different stable configurations, made up of a series of hinged triangles. Certain embodiments relate to such polyhedrons which can be moved to form six, or more, configurations. Other embodiments involve triangles having all three sides unequal. Other embodiments involve biased hinges made of cardboard having a heat-set thermoplastic polymer layer secured thereto.

United States Patent 1191 Hooker 1451 July 15,1975

1 1 POLYHEDRAL ANNULAR STRUCTURES AND BLANKS FOR FORMING SAME [76] Inventor: Rea Ferdinand Hooker, 170 W. 74th St., New York, N.Y. 10023 [22] Filed: Apr. 27, 1973 [21] Appl. No.: 355,037

3.302.321 2/1967 Walker 46/1 3,346,998 10/1967 Nelson 46/21 x FOREIGN PATENTS OR APPLICATIONS 653.204 ll/l962 Canada 52/86 Primary Examiner-F. Barry Shay Attorney, Agent, or FirmAbner Sheffer [57] ABSTRACT Folded structures which are polyhedrons of generally toroidal shape, movable to various different stable configurations, made up of a series of hinged triangles. Certain embodiments relate to such polyhedrons which can be moved to form six, or more, configurations. Other embodiments involve triangles having all three sides unequal. Other embodiments involve biased hinges made of cardboard having a heat-set thermoplastic polymer layer secured thereto.

17 Claims, 56 Drawing Figures /l/'/1 3 1 1's}; 1 a; .3 I1tt.""' 11111d11 11 1 1 Ct fin "we/Q11 511 116 5 1 gal w [312111110 POLYHEDRAL ANNULAR STRUCTURES AND BLANKS FOR FORMING SAME This invention relates to a polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a contiguous multiplanar toroidal web having two edges, which structure can be rotated, about its core, into various different stable configurations each of which is radially substantially symmetrical about said central axis.

Walker US. Pat. No. 3,302,321 describes a folded structure which is a polyhedron of generally hexagonal appearance composed of three rows of interior isosceles right triangles and two rows of edge isosceles triangles. The folded structure can be rotated to make five different stable configurations.

One embodiment of this invention relates to folded structures which can be rotated to make six, seven or more different stable configurations. Other embodiments of this Invention relate to folded structures which can be rotated to make a plurality of different stable configurations of differing heights. Still other embodiments involve biased hinges made of cardboard having a heat-set thermoplastic polymer layer secured thereto.

Several forms of the invention are illustrated in the accompanying drawings in which FIG. I is a plan view of a blank for constructing a polyhedron consisting of six rings of contiguously hinged triangles.

FIGS. 1A through IG are plan views of portions of modified blanks as in FIG. 1, FIGS. 1A through IE illustrating changes with respect to the edge rings of triangles.

FIGS. 2 is a plan view of the flexible tubular structure, before final assembly, produced by folding the blank of FIG. 1.

FIG. 2A is another view of the folded tubular structure. showing a portion not visible in FIG. 2.

FIG. 2B is a plan view of a block produced by compressing the tubular structure (of FIGS. 2 and 2A) endto-end.

FIGS. 3 through 20 illustrate the different positions the assembled polyhedron will assume through a rotating" or traversing action.

FIGS. 3, 9, and are top, side, and bottom views respectively of the assembled polyhedron in a first position, hereafter termed also a solid position.

FIGS. 4, l0, and 16 are top. side, and bottom views respectively ofa second position into which the polyhedron is transformed.

FIGS. 5, 11, and 17 are top, side, and bottom views respectively of a third position into which the polyhedron is transformed.

FIGS. 6, l2, and 18 are top, side, and bottom views respectively of a fourth position into which the polyhedron is transformed.

FIGS. 7, 13, and 19 are top, side, and bottom views respectively of a fifth position into which the polyhedron is transformed.

FIGS. 8, 14, and 20 are top, side, and bottom views respectively of a sixth position into which the polyhedron is transformed.

FIGS. 21 to 24 illustrate the formation of a polyhedron having seven stable configurations. FIG. 21 is a plan view of a blank. FIGS. 22 and 23 are top and side views of the polyhedron in a solid configuration, and

FIG. 24 is a side view of the same polyhedron transformed to another configuration.

FIGS. 25-27 illustrate the formation of still another polyhedron having seven stable configurations, two of which are solid. FIG. 25 shows the blank in plan view and FIGS. 26 and 2 7 are top and side views, respectively, of two of the stable configurations.

FIGS. 28-44 illustrate the formation of polyhedrons comprising non-isosceles triangles.

FIG. 28 is a plan view of a blank for constructing a polyhedron consisting of five rings of triangles; FIGS. 29, 30 and 31 are top, side and bottom views of that polyhedron in solid position and FIGS. 32 and 33 are top views of two other positions. FIG. 28A illustrates the relationship of the edge triangles.

FIG. 34 is a plan view of a portion of a blank for constructing another polyhedron consisting of five rings of triangles, including three rings of triangles of grossly unequal sides; FIGS. 35, 36 and 37 are top, side and bottom views of that polyhedron in solid position.

FIG. 38 is a plan view ofa portion of a blank for constructing a polyhedron consisting of six rings of triangles; FIGS. 39, 40 and 41 are top, side and bottom views of that polyhedron in solid position; FIG. 38A illustrates the relationship of the edge triangles.

FIG. 42 is a plan view ofa portion of a blank for constructing another polyhedron consisting of six rings of triangles. FIG. 43 is a bottom view of that polyhedron when it has been rotated to its third position and FIG. 44 is a side view of the polyhedron when it has been rotated to its fourth position. FIG. 42A illustrates the relationship of the edge triangles.

The blank shown in FIG. 1 may be used to make a rotatable polyhedron which is of generally nonagonal appearance More specifically, in its stable configurations and in plan view (viewed from the top, or bottom, when the axis of rotation is vertical), it exhibits an outline which can be followed, along the material and without a break, through nine outer points which lie approximately on the circumference of an (imaginary) horizontal circle having the axis of rotation of the polyhedron at its center. These various outlines can be seen in the top views of successive configurations shown in FIGS. 3, 4, 5, 6, 7 and 8 and in the corresponding bottom views shown in FIGS. 15, 16, I7, l8, l9 and 20, respectively.

The rectangular blank A shown in FIG. 1 has fold lines l1, l2 and 13 which define six *horizontal rows of triangles (the triangles of each row being designated as I, 2, 3, 4, 5 and 6 respectively) arranged in eighteen vertical" files. The triangles of each row have, alternately, a common side (such as side B which is common to two triangles of the second row) or a common apex (such as point C which is common to two triangles of that second row). Each successive pair of vertical fold lines 11 defines a vertical file (of the six triangles) with the common sides (e.g. B and the common apices (cg. C being situated on the lines 11. The other fold lines 12 and 13 run through the common apices and constitute the other two sides of each triangle. All fold lines 11 are parallel to each other and spaced equally, as are all fold lines 12 and all fold lines 13.

In the configuration shown in FIGS. 1-24 all the triangles 2, 3, 4 and 5 are congruent obtuse isosceles triangles, the angle D" at the obtuse apex of each triangle being about 108 (and the other two angles of the triangle therefore being about 36 each).

In addition to the triangles l, 2, 3, 4, 5, 6 the blank A also has, at one end, three attaching tabs 7, 8, 9 which are designed to be secured (e.g. by suitable adhesive) to the triangles 2b, 4b, 6b, respectively, at the other end of the blank during the assembly of the device.

All the vertical fold lines 11 are infold lines; that is, they are to be folded to bring together the faces of the two triangles of a given row which have common sides along the fold line. Thus, line ll near the left side of FIG. 1 is folded so as to bring the faces of the two triangles 3 and 3" toward each other; see FIG. 2.

All the diagonal fold lines 12, 13 are outfold lines. Again see FIG. 2.

It should be understood that a fold line which is described as an infold" line, when viewed from one face of the blank, can be described as an outfold line when it is viewed from the opposite face. The words infold" and outfold" are here used with reference to that face of the blank which will form the outside of the final folded configuration; see FIG. 9-14 for instance. That outside face can be termed the obverse" face as compared to the face forming the interior or reverse face.

When the blank A is folded as described above, it forms a flexible structure as shown in FIG. 2. Such folding causes the top and bottom edges E and F of the blank to approach each other so that the structure shown in FIG. 2 is tube-like, almost completely enclosing a central aperture. See FIG. 2A. This tube-like structure can be compressed end-to-end (as shown in FIG. 28) to form a pentagonal block having a central pentagonal passage G with the edges E and F touching each other. In this pentagonal block all the triangles of each particular row are superimposed one on top of the other. As seen in FIG. 2B, the obverse faces of end triangles 2b, 4b and 6b lie in the top plane of the compressed block. Just below them there are visible parts of the reverse faces Ir, 3r and Sr, of triangles l, 3 and 5, respectively.

The flexible folded structure shown in FIG. 2 can be shaped into circular form so that the left hand end des ignated 11a is superimposed on the infold line at the opposite end, with the tabs 7, 8 and 9 being adhered to the backs of the end triangles 2b, 4b, and 6!), respectively, and the line 11a becomes substantially identical to all the other vertical infold lines 11. The resulting structure has the configuration shown in FIGS. 3, 9 and 15. All the triangles 2, 3, 4 and 5 are exposed while all the triangles l and 6 are substantially hidden by substantially complete infolding of those portions of lines 11 which constitute the boundaries of such triangles. In this position certain points are situated approximately at the vertical axis (y in FIG. 3). These axial points are: (a) the points of contact of triangles 2 and top edge E, which contact points are visible from the top (FIG. 3) and are situated above mid-height of the structure; and (b) the points of contact of triangles S and bottom edge F, which contact points are visible from the bottom (FIG. 15) and are situated below the mid-height of the structure.

The structure can be rotated by pushing it inward at the top and pulling it outward at the bottom. It is thus brought to the position shown in FIGS. 4, 10 and 16 in which triangles 3, 4, 5 and 6 are visible while all the triangles 1 and 2 are substantially hidden by substantially complete infolding of those portions of lines 11 which constitute the boundaries of such triangles. In this position again there are two sets of points which are situated at about the vertical axis. One set of these axial points is the same as set (a) mentioned in the preceding paragraph, but in this position those points are below the mid-height of the structure. The other set of these axial points is (c) the obtuse apices of triangles 2, which points are now above the mid-height of the structure. In addition the outfold lines joining these two sets of points are located substantially axially, these lines being those portions of outfold lines 12 and 13 which constitute the boundaries between the row of triangles 1 and the row of triangles 2.

The structure can next be rotated, again by pushing it inward at the top and pulling it outward at the bottom. It is thus brought to the position shown in FIGS. 5, 11 and 17 in which triangles 4, S, 6, and 1 are visible while all the triangles 2 and 3 are substantially hidden by substantially complete infolding of those portions of line 11 which constitute the boundaries of such triangles. Here again there are two sets of points, and the outfold lines joining them, which are situated approximately at the vertical axis. The set (c) mentioned in the preceding paragraph is now below the mid-height while the obtuse apices of triangles 3 are above the midheight, as are the lines joining the obtuse apices of triangles 2 and 3.

The same process can be repeated three more times. In each position, two rows of triangles are substantially hidden by infolding and two sets of apices of triangles are located substantially axially. Various portions of the interior (that is the reverse face of the blank) will be visible in certain positions; these interior portions are indicated by shading in FIGS. 6, 7, 8, I6, 17 and 18. It will be noted that, of the six illustrated positions, there in one (shown in FIGS. 3, 9 and 15) in which the interior is completely hidden; this may be termed the solid" position since it in itself gives no indication that the polyhedron is really hollow.

During the process of rotating from one stable position to another, the structure passes, of course, through a whole series of intermediate positions. If the rotating forces are removed when the structure is in an intermediate position it tends to return to the preceding stable position or to move, by itself, to the next stable position. For instance in the process of rotating the structure from its solid" position (FIG. 9) to the next position (FIG. 10) certain stresses are induced. Among these are stresses at the infold lines 11 of the row of triangles 4, which stresses arise because the infold lines I] are creases in the material (which tends to keep its pre-set creased condition) and the rotation (at first) causes the triangles 4 to move further from the axis and thus tends to straighten out those creases. Stresses are also induced at other foldlines, such as the infold lines 11 of the row of triangles 2 in that the rotation causes those infold lines to be folded past their pre-set creased condition. After the rotating step has proceeded part way between stable positions, to a position at which the total stress may be at its highest, further rotation causes the balance of forces to change and the structure tends to snap (like a toggle joint) into its next stable position. It will be understood that the fold lines need not be creases but can be other types of hinges, pre-set to have the desired bias. For instance the triangles may be of individual pieces of rigid plastic which are joined together (at the fold lines") by spring-loaded hinges which may themselves be thin plastic strips premolded to a V cross section and adhered to the separate triangles to form the desired infold and outfold lines, or the blank may be shaped from a sheet of plastic, the infold and outfold lines being formed by heating at the fold line, bending that line in the desired direction to the desired extent and then cooling, while so bent, to set the material in that condition. In one suitable procedure a cardboard blank having a thermoplastic layer laminated thereto (such as a thin transparent film of known polyvinyl chloride type) is folded into the collapsed shape shown in FIG. 2 (or 2B), heated, e.g. to a temperature (such as 180F) at which the stresses in the plastic layer resulting from such folding are largely relieved (but preferably below the temperature at which it becomes tacky) and then cooled to room temperature while in the collapsed shape, thus setting the plastic in its bent configuration. The material at the fold lines need not be one which, like cardboard, is substantially inextensible under the conditions of use. Instead, at some (or all) of the fold lines it may be of a material such as a stable synthetic rubber (e.g. elastomeric polyurethane) which is reversibly elastic under the conditions of use; the employment of such material permits the outermost portions of the final structure to expand during its movement from one stable position to the next and permits one to produce rotatable structures that would otherwise be incapable of full rotation.

The thermoplastic polymer layer mentioned above may be applied as a film or as a relatively stiff sheet (e.g. -15 mils thick) or as a coating. Particularly when the polymer layer is relatively stiff it is preferably prescored, as by pressure exerted locally on the surface of the laminate at the fold lines to locally decrease the thickness of the film and/or of the cardboard (or other fibrous structure), to facilitate subsequent folding. The laminate may comprise layers of plastic on both sides of the cardboard, and may be scored on both sides. Also, particularly when the polymer layer is relatively stiff, that layer preferably does not cover the entire surface, but has cutouts at the areas corresponding to all the apices of the triangles; for example, in a structure in which the equal sides of the isosceles triangles are about 3 inches long, the plastic sheet may have a series of circular cut-outs (holes) each about A inch in diameter centered on the points corresponding to the apices of the triangles, to facilitate folding at said apices without unduly stressing the plastic layer. A plasticlaminated structure has a better quality surface appearance and is longer lasting in use.

In FIG. 21 there is shown a blank H for constructing a polyhedron with seven stable configurations which is of generally regular tetradecagonal appearance. In its stable configurations in plan view it has an outline which can be followed, along the material and without a break, through fourteen outer points which lie approximately on the circumference of an imaginary horizontal circle having the axis of rotation of the polyhedron at its center. The seven rows of triangles l4, l5, l6, l7, l8, I9, are defined by infold lines 11 and outfold lines 12, 13, and are arranged in 28 filed bounded by the lines 11. The obtuse angle din this case is 120; as in FIG. 1, there are tabs at one end for use in assembling the polyhedron; these tabs lSY, 17Y, 19Y are adapted to be adhered to the triangles 15, 17, I9, respectively, at the other end of the blank. FIGS. 22 and 23 are top and side views of the resulting assembled polyhedron. Here again in each stable position two rows of triangles (i.e. triangles l4 and 20, in FIGS. 22 and 23) are substantially hidden, and the other triangles are visible, and two sets of apices lie approximately on the axis of the polyhedron in these stable positions. FIG. 24 is a side view of the same polyhedron rotated to another position. When the blank is folded and collapsed fully (like the collapsed blank of FIG. 1 shown in FIG. 28) it forms a hexagonal block having a central hexagonal passage.

Other structures having still more rows of triangles and still more stable positions are made in the same way, with the following parameters in which the first number in each case is the number of rows of triangles, followed by the approximate number of triangles per row, followed by the approximate angle d; 8, 38, l28.6;9,48, 10, 58,140"; ll,68, I44"; I2, 78, 147.3; I3, 88,

FIG. 25 shows a blank Q for constructing another polyhedron with seven stable configurations, but this polyhedron is of generally regular dodecagonal appean ance. It differs from the blank H of FIG. 21 in that each of the seven rows of triangles (14, I5, 16, l7, 18, 19, 20) has 24 rather than 28 triangles per row and the obtuse angle D is 114. FIG. 26 shows the resulting polyhedron in the solid position. But this particular design also has another solid position, in which the top and bottom triangles meet. This is shown in FIG. 27, which should be compared to the corresponding position, shown in FIG. 24, for the polyhedron made with 28 triangles in each of seven rows. Other structures providing two *solid" positions are as follows, in which the first number in each case is the number of rows of triangles, followed by the approximate number of triangles per row, followed by the approximate angle d: 9, 44, 131.8"; 11, 64, 142; 13, 84, 148.7". When the number of rows is even, a similar type of structure is formed except that, although both edges meet and are located in substantially the same horizontal plane transverse to the axis, the triangles at the two edges are not in phase; examples of such structures are (using the same system as above) 8, 34, 124.3"; I0, 54, I375"; 12, 74, l45.7.

The main triangles need not be isosceles triangles. In the embodiment shown in FIGS. 38-41 there are (as in the embodiment of FIGS. l-24) four horizontal rows of main triangles 34, 35, 36, 37 plus two horizontal" rows of edge triangles 33, 38, all arranged in vertical" files, defined by straight vertical infold lines 11, and all the main triangles are congruent. In the embodiment of FIGS. l-24 the sides which each main triangle has in common with a triangle of the next row (i.e. the sides that are outfold lines) are of equal length. But in the embodiment shown in FIGS. 38-41 (as well as in those shown in FIGS. 28-37 and 42-44) those sides are not equal, side 12S (FIG. 38) being shorter than side 13L and side 135 being shorter than side 12L. (In the illustrated embodiments all sides 128 are parallel, as are all sides 12L, all sides 13S, and all sides 13L; sides 13L and 121. are of equal length; sides 13S and 128 are of equal length). As will be seen from FIGS. 38 and 41 there are 20 triangles in each horizontal row and the resulting rotatable polyhedron is of generally decagonal configuration. It has six stable positions. It will be noted that the triangle resulting from juxtaposition of edge triangles 33, 38 (see FIG. 38A) is smaller than a triangle 34, 35, 36 or 37. Angle D" (at the obtuse apex of each main triangle) is 108.

The embodiment shown in FIGS. 42-44 also is made up of non-isosceles main triangles; e.g. side 12s and 13s are shorter than sides 12], I31. Here again there are an even number (four) horizontal rows of congruent main triangles (42, 43, 44, 45). The triangle resulting from the juxtaposition of edge triangles 41 and 46 (see FIG. 42A) is larger than a main triangle; the number of triangles per horizontal row is 22, which is more than in FIGS. 38-41, and the angle "D is larger, i.e. 114.

In some of the embodiments, such as some of those using non-isosceles main triangles, the polyhedron has stable positions in which the central portions are spaced a considerable distance from the axis; that is, the portions corresponding to the axial points" (of the embodiment of FIGS. 1-24) discussed previously are situated at a substantially uniform distance from the axis itself, leaving the central portion of the polyhedron open. The diameter of the central circular opening may change as the polyhedron is rotated.

The concept of using non-isosceles triangles may also be applied to structures which, like that shown by Walker, have only three rows of main triangles. In the embodiment shown in FIGS. 28, 29, 30 and 31 the three rows of main triangles 22, 23, 24 are again arranged in vertical files, defined by straight vertical infold lines 11; all the main triangles are congruent, but all have unequal sides, at their two outfold line boundaries. There are twelve triangles in each row. Angle D" is 90. The embodiment shown in FIGS. 34-37 is similar (having three rows of main triangles 29, 30, 31) but the inequality of the sides of these triangles is greater, the number of triangles in each row is greater (there being 20 triangles per row giving a decagonal polyhedron); the angle D is 90. In both these embodiments (each having an odd number of rows) the trian gle formed by juxtaposing the edge triangles (21, 25 or 28, 32) is equal to one of the main triangles (see FIG. 28A) whereas in the illustrated non-isosceles embodiments having an even number of rows the triangle formed by such juxtaposition is not equal to a main triangle. It will be understood that the number of horizontal rows may be greater than 3 or 4, e.g., 5, 6, or 7, with corresponding adjustments in the number of triangles per row to permit rotation.

The structures may be modified in various ways. For instance, the triangles of the edge rows (such as edge triangles 1 and 6 of FIGS. 1-24, which triangles correspond to bisected triangles 2, 3, 4 or 5) may be altered in shape; see for instance the edge triangles 6A, 6B illustrated in FIGS. IA and 18. Also the edge triangles may have only two straight sides, the third side being curved or crooked (as illustrated by edge triangles 6C and 6D in FIGS. IC and 1D) or one row of such triangles (say triangles 1) may be removed entirely and the other set enlarged to a size and shape equal (or almost equal) to that of the main triangles as illustrated by edge triangles GB in FIG. IE; or both rows of edge triangles may even be omitted. The main triangles (and edge triangles) may have apertures or windows, as indicated by W in FIG. 16.

The blank need not be collapsed (e.g. to the position shown in FIG. 2) before its ends are joined together; instead the ends may be joined first, forming a sort of cylinder, after which the folding may be effected; thus a blank of flexible metal may be formed into a cylinder,

and then formed into folded configuration with a mold. The blanks need not be rectangular; they may, for instance, be parallelograms, adapted to be joined together at their diagonal edges. FIG. 1F shows such a modification of the blank A of FIG. 1 (only the two end portions of the parallelogram are shown, the intermediate center portion being omitted), the right hand and having attaching tabs 1T, 2T, 3T, 4T, 5T located at its diagonal edge and adapted to be adhered, during the assembling of the device, to triangles 1, 2, 3, 4, 5 respectively (at the diagonal edge 13a at the left hand end), as by means of a moisture-activatable adhesive layer on the outside face of the tabs. The blank need not be a single element; that is, instead of using a single large blank several smaller suitable scored sheets (e.g. two sheets each having half the total number of triangles) may be employed as the blank, being joined together at their edges at any stage in the assembly.

The articles described herein have various uses, e.g., as amusement devices, as geometrical education devices, for advertising or display purposes, as housings (when partly or wholly transparent or windowed) for lights, etc. Different triangles may be differently colored or textured to provide various ornamental effects.

It will be evident that the structures, while polyhedral, may be considered to have, roughly, the general configuration of a torus (or doughnut); in the configuration shown in FIG. 3-20 the diameter of the hole in the middle of the torus is zero, while in FIGS. 22 and 26 there is a small hole in the middle. The folded triangles form a continuous multiplanar toroidal web having two edges, i.e., the edges visible in FIG. 12. Rotation" occurs about the core of the toroidal structure (for a true torus, the core" is the imaginary circular hori zontal line encircling the central rectilinear vertical axis and joining the centers of all the vertical circular cross sections of the torus).

It is understood that the foregoing detailed description is given merely by way of illustration and that variations may be made therein without departing from the spirit of the invention. The Abstract" given above is merely for the convenience of technical searchers and is not to be given any weight with respect to the scope of the invention.

I claim:

1. A polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges, which structure can be rotated, about its core, into at least six different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising congruent plane triangles arranged in at least four rows of adjacent triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common being outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the apical angles of said triangles being at least about 108, each of said rows having at least 18 triangles, the structure, when in each of said stable configurations, having two sets of axially spaced apical points situated approximately at said central axis.

2. A structure as in claim 1 in which two of said rows are adjacent to said edges of said structure and the others of said rows are arranged between said edgeadjacent rows, the triangles of said edge-adjacent rows each have a side which is not common to another triangle of said rows, and said non-common sides of adjacent triangles of said rows are connected by a folded web having an infold hinge in one of said radial planes, with outfold hinges at said non-common sides.

3. A structure as in claim 1 in which there are only four of said rows, said structure being rotatable to only six different stable positions, said apical angle being about 108.

4. A structure as in claim 1 in which there are five of said rows, said structure being rotatable to seven different stable positions, said apical angle being at least about 114.

5. A structure as in claim 1 in which said congruent triangles are isosceles triangles.

6. A structure as in claim 1 in which the three sides of each of said triangles are unequal.

7. A structure as in claim 1 in which said hinges are integral with said triangles, said hinges and triangles being of foldable sheet material.

8. A structure as in claim 2 in which each of said folded webs comprises a pair of equal triangles and said structure is a nonagonal polyhedron and is which there are only four of said rows, said apical angle is about 108, said structure being rotatable to only six said different stable positions, and each of said rows has only ]8 triangles.

9. A structure as in claim 3 in which each of said rows has only 18 triangles.

10. A polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges which structure can be rotated, about its core, into at least five different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising at least three rows of adjacent congruent plane triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges, arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common being outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the three sides of each of said triangles being unequal.

l]. A structure as in claim 10 in which said hinges are integral with said triangles, said hinges and triangles being of foldable sheet material.

12. A polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges, which structure can be rotated, about its core, into at least five different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising at least three rows of adjacent congruent plane triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a rows being infold hinges arranged within planes which radiate from and include said axis and the sides which adjacent row have in common being outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, said structure being of cardboard, said cardboard having, at least at said hinges thereof, an upper layer of thermoplastic polymer which has been heat set to bias said hinges in the folded direction.

13. A structure as in claim 12 in which said thermoplastic layer is up to 15 mils thick.

14. A structure as in claim 12 in which said thermoplastic layer is a polyvinyl chloride film.

15. A flat blank of sheet material, said blank having means, including score lines at which said blank can be folded and edges adapted to be secured together to form a continuous web after said blank is folded at said score lines, for converting said blank to a polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges, which structure can be rotated, about its core, into at least six different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising congruent plane triangles arranged in at least four rows of adjacent triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common being outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the apical angles of said triangles being at least 108, each of said rows having at least 18 triangles, the structure, when in each of said stable configurations, having two sets of axially spaced apical points situated approximately at said central axis. said score lines being situated so that, on folding, they form the boundaries of said triangles.

16. A flat blank of sheet material, said blank having means, including score lines at which said blank can be folded and edges adapted to be secured together to form a continuous web after said blank is folded at said score lines, for converting said blank to a polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges which structure can be rotated, about its core, into at least five different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising at least three rows of adjacent congruent plane triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges, arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common beeing outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the three sides of each of said triangles being unequal, said score lines being situated so that, on folding. they form the boundaries of said tri- 12 angles.

17. A blank as in claim 15, said polyhedral structure being a nonagonal polyhedron in which two of said rows are adjacent to said edges of said structure and the others of said rows are arranged between said edgeadjacent rows, the triangles of said edge-adjacent rows each have a side which is not common to another triangle of said rows, and said non-common sides of adjacent triangles of said rows are connected by a folded web having an infold hinge in one of said radial planes, with outfold hinges at said non-common sides, each of said folded webs comprises a pair of equal triangles, there are only four of said rows, said apical angle is about 108, said structure being rotatable to only six said different stable positions, and each of said rows has only 18 triangles.

UNITED STATES PATENT OFFICE CERTIFICATE OF CORRECTION Patent No. 3 8914352 D te July lg 1975 I t Rea Ferdinand Hooker It is certified that error appears in the above-identified patent and that said Letters Patent are hereby corrected as shown below:

Column 6, line 18, "88, should be ---88,150.---

Column 10, line 12, "within 1: rows" should be---w1thin a row-- Signcd and Scaled this n wwh Day Of July 1980 [SEA L| Arrest:

SIDNEY A. DIAMOND Arresting Oflicer Commissioner of Patents and Trademarks 

1. A polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges, which structure can be rotated, about its core, into at least six different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising congruent plane triangles arranged in at least four rows of adjacent triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common being outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the apical angles of said triangles being at least about 108*, each of said rows having at least 18 triangles, the structure, when in each of said stable configurations, having two sets of axially spaced apical points situated approximately at said central axis.
 2. A structure as in claim 1 in which two of said rows are adjacent to said edges of said structure and the others of said rows are arranged between said edge-adjacent rows, the triangles of said edge-adjacent rows each have a side which is not common to another triangle of said rows, and said non-common sides of adjacent triangles of said rows are connected by a folded web having an infold hinge in one of said radial planes, with outfold hinges at said non-common sides.
 3. A structure as in claim 1 in which there are only four of said rows, said structure being rotatable to only six different stable positions, said apical angle being about 108* .
 4. A structure as in claim 1 in which there are five of said rows, said structure being rotatable to seven different stable positions, said apical angle being at least about 114* .
 5. A structure as in claim 1 in which said congruent triangles are isosceles triangles.
 6. A structure as in claim 1 in which the three sides of each of said triangles are unequal.
 7. A structure as in claim 1 in which said hinges are integral with said triangles, said hinges and triangles being of foldable sheet material.
 8. A structure as in claim 2 in which each of said folded webs comprises a pair of equal triangles and said structure is a nonagonal polyhedron and is which there are only four of said rows, said apical angle is about 108*, said structure being rotatable to only six said different stable positions, and each of said rows has only 18 triangles.
 9. A structure as in claim 3 in which each of said rows has only 18 triangles.
 10. A polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges which structure can be rotated, about its core, into at least five different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising at least three rows of adjacent congruent plane triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges, arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common being outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the three sides of each of said triangles being unequal.
 11. A structure as in claim 10 in which said hinges are integral with said triangles, said hinges and triangles being of foldable sheet material.
 12. A polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges, which structure can be rotated, about its core, into at least five different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising at least three rows of adjacent congruent plane triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a rows being infold hinges arranged within planes which radiate from and include said axis and the sides which adjacent row have in common being outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, said structure being of cardboard, said cardboard having, at least at said hinges thereof, an upper layer of thermoplastic polymer which has been heat set to bias said hinges in the folded direction.
 13. A structure as in claim 12 in which said thermoplastic layer is up to 15 mils thick.
 14. A structure as in claim 12 in which said thermoplastic layer is a polyvinyl chloride film.
 15. A flat blank of sheet material, said blank having means, including score lines at which said blank can be folded and edges adapted to be secured together to form a continuous web after said blank is folded at said score lines, for converting said blank to a polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges, which structure can be rotated, about its core, into at least six different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising congruent plane triangles arranged in at least four rows of adjacent triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common being Outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the apical angles of said triangles being at least 108*, each of said rows having at least 18 triangles, the structure, when in each of said stable configurations, having two sets of axially spaced apical points situated approximately at said central axis, said score lines being situated so that, on folding, they form the boundaries of said triangles.
 16. A flat blank of sheet material, said blank having means, including score lines at which said blank can be folded and edges adapted to be secured together to form a continuous web after said blank is folded at said score lines, for converting said blank to a polyhedral structure which is radially substantially symmetrical about a central axis, said structure comprising a number of planar triangles hinged together at their sides so as to form a continuous multiplanar toroidal web having two edges which structure can be rotated, about its core, into at least five different stable configurations each of which is radially substantially symmetrical about said central axis, said structure comprising at least three rows of adjacent congruent plane triangles, each of said rows being a ring of single triangles arranged in alternation so that each triangle has a hinged side in common with one of its two neighbors of its row and has an apical point in common with the other of its neighbors of its row, said rows interfitting so that each triangle of each row has a side in common with a triangle of the adjacent row, the common sides within a row being infold hinges, arranged within planes which radiate from and include said axis and the sides which adjacent rows have in common beeing outfold hinges, said apical points being situated on the intersections of infold and outfold hinges, the three sides of each of said triangles being unequal, said score lines being situated so that, on folding, they form the boundaries of said triangles.
 17. A blank as in claim 15, said polyhedral structure being a nonagonal polyhedron in which two of said rows are adjacent to said edges of said structure and the others of said rows are arranged between said edge-adjacent rows, the triangles of said edge-adjacent rows each have a side which is not common to another triangle of said rows, and said non-common sides of adjacent triangles of said rows are connected by a folded web having an infold hinge in one of said radial planes, with outfold hinges at said non-common sides, each of said folded webs comprises a pair of equal triangles, there are only four of said rows, said apical angle is about 108said structure being rotatable to only six said different stable positions, and each of said rows has only 18 triangles. 